Phase Behavior and Densities of Propylene+ Hexane Binary Mixtures

Article pubs.acs.org/jced

Phase Behavior and Densities of Propylene + Hexane Binary Mixtures to 585 K and 70 MPa Rajendar R. Mallepally,*,† Venkat S. Gadepalli,† Babatunde A. Bamgbade,† Nathaniel Cain,‡ and Mark A. McHugh† †

Department of Chemical and Life Science Engineering, Virginia Commonwealth University, 601 West Main Street, Richmond, Virginia 23284, United States ‡ Afton Chemical Corporation, 500 Spring Street, Richmond, Virginia 23219, United States S Supporting Information *

ABSTRACT: In this study, we report phase behavior data for propylene + hexane mixtures at temperatures of 295 to 468 K and pressures to 5.5 MPa and high-pressure mixture density data at temperatures of 295 to 584 K and pressures to 70 MPa. Both the phase behavior and density data are simultaneously determined using a variable volume, high-pressure view cell that is coupled with a linear variable differential transformer. The phase behavior and mixture density data are modeled with the Soave−Redlich−Kwong (SRK), Peng−Robinson (PR), modified Sanchez−Lacombe (MSL), and perturbed-chain statistical associating fluid theory (PC-SAFT) equations of state (EoS). The PC-SAFT and MSL EoS provide the best fit of the phase behavior data with a nonzero value of 0.028 for kij. Likewise, the PC-SAFT EoS provides the best fit of the high-pressure mixture density data, though the PC-SAFT equation slightly overpredicts the solution density and the calculated densities are relatively insensitive to changes in kij from zero to 0.028.

1. INTRODUCTION High-pressure and high-temperature phase behavior and fluid property data are important for the design, optimization, and control of chemical and petrochemical processes.1 Phase boundary curves are needed to know when a process is operated in a multiple-phase region. In this study, vapor−liquid equilibrium (VLE) data for the propylene + hexane system are reported at high temperatures, from 295 to 468 K, and pressures up to 5.5 MPa. The present study complements previously reported studies reported on the phase behavior of ethylene + hexane mixtures.1,2 The VLE data reported here also extend the available database for the propylene + hexane system, which is limited to temperatures below 333 K and pressures below 1.0 MPa.3 In addition to VLE data, mixture density data provide important thermophysical property information needed for the design of transport and processing facilities.4 To the best of our knowledge, there are no high-temperature, high-pressure density data available in the literature for propylene + hexane mixtures. Here, we report mixture density data from 295 to 584 K and pressures to 70 MPa for four propylene + hexane compositions. We also present the capabilities of several contemporary equations of state (EoS) to model propylene + hexane phase behavior and densities over the entire experimental temperature, pressure, and composition range employed in this study. The EoS used here include two cubic equations, the Soave−Redlich−Kwong (SRK)5 and the Peng−Robinson (PR),6 a lattice-gas equation, the modified Sanchez−Lacombe (MSL),7,8 and a perturbation equation, the perturbed-chain statistical associating fluid theory (PC-SAFT).9 The density data reported here are also correlated © XXXX American Chemical Society

with the modified Tait equation as a function of temperature and pressure for each investigated composition to facilitate the calculation of mixture density for a wide range of conditions.

2. MATERIALS AND METHODS 2.1. Materials. Table 1 lists the mole fraction purities of the chemicals used in this study. Both propylene and hexane are purchased from Sigma-Aldrich Corporation and used as received. Table 1. Purity of the Compounds Used in This Study chemical name

source

mole fraction purity

analysis method

propylene hexane

Sigma-Aldrich Sigma-Aldrich

1.000 0.9783

GCa GC−MSb

a

Gas chromatography (GC) as stated by the supplier. bDetermined in this study using GC−MS and impurities information provided in the Supporting Information

2.2. Methods. 2.2.1. General. Figure 1 shows the schematic diagram of the view cell used for VLE and density experiments. Found elsewhere10−12 are more details on this apparatus. In the present case, the view cell uses a metal bellow rather than an O-ring-sealed piston to generate pressure. A thin Kapton film is Special Issue: In Honor of Kenneth R. Hall Received: February 29, 2016 Accepted: June 13, 2016

A

DOI: 10.1021/acs.jced.6b00181 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

Figure 1. Schematic diagram of the high-pressure view cell used in this study for phase behavior and density measurements.

when the cylinder is disconnected from the cell. The final mass of propylene in the cell includes the amount loaded with the syringe and the amount in the cell after flushing. The cell is then heated and pressurized to the desired conditions where both VLE and density data are recorded simultaneously. 2.2.2. Phase Equilibrium and Density Measurements. Once the desired temperature and pressure is reached, the fixed composition mixture is stirred vigorously for 30 min to ensure thermal equilibrium. At a constant temperature, with a clear, single phase in the view cell, the system pressure is slowly reduced by ∼0.5 MPa and the mixture again is stirred vigorously and allowed to come to thermal equilibrium. If a clear, single phase exists at this lower pressure, the system pressure is again slowly decreased by ∼0.5 MPa and the mixture again is stirred vigorously and allowed to come to thermal equilibrium. This pressure reduction technique is continued until either a small vapor bubble (bubble point, BP) appears or a mist or dew appears (dew point, DP). The pressure is now increased well into the single phase region and the solution is mixed to return to equilibrium. The pressure reduction technique is repeated with smaller step changes in pressure until the reported transition pressure is within an observed 0.07 MPa interval between the clear, single phase and the BP or DP. For both types of transitions, the composition of the predominant phase is considered equal to the overall solution composition as the amount of mass present in the second phase is negligible. Critical points (CP) are obtained by the same pressure reduction technique although now the mixture exhibits a light orangish critical opalescence and the single phase solution separates into equal-sized liquid and vapor volumes. The reproducibility of BP and DP measurements is determined by repeating measurements at a minimum of two temperatures for each isopleth. Data are obtained at random pressures for any given isopleth to minimize any potential experimental artifacts in the measurements. Isothermal mixture density data are recorded in the singlephase region from the BP pressure to ∼70 MPa. For each isotherm, the pressures are chosen in random order to minimize any experimental artifacts in the measurements. The expanded uncertainty15 of the mole fractions is 0.0007 for both VLE and density measurements. The standard uncertainties of temperature and pressure for both types of experiments

used to seal the window against the base of the window holder that is inserted in the front of the cell where the holder makes a metal-to-metal seal. Hence, elastomeric O-rings previously used by our group10−13 to seal the piston and sapphire window are eliminated with these design changes, which improves the reliability of the apparatus when operating at high temperatures. The system pressure, generated by introducing water into the bellows, is measured with a Heise pressure gauge (Heise Corporation, Model CM57303, 0−68.9 MPa, standard uncertainty of 0.07 MPa). The pressure needed to extend or compress the bellow varies linearly up to 0.15 MPa and is accounted for in the data reported here. The temperature is measured using a type-k thermocouple (Omega Engineering) calibrated against an accurate RTD thermometer having certificate of calibration traceable to NIST standards (H−B Instrument company, Model DURAC TP-R04, measurement range 173 to 673 K, permissible deviation is °C = ±[0.15 + 0.002·t], where t is the temperature in °C). The maximum uncertainty ranges from 0.2 K at the lowest temperature investigated in this study to 0.8 K at highest temperature investigated in this study. However, in this study the temperature is controlled to within 0.2 K for both VLE and density measurements. The internal cell volume is determined using a linear variable differential transformer (LVDT, Schaevitz Corporation, Model 2000 HR) that tracks a magnetic core at the end of a rod connected to the front, internal face of the bellow as shown in Figure 1. The internal cell volume, as a function of LVDT reading, is calibrated using accurate hexane density data obtained from the NIST Chemistry WebBook.14 The contents of the cell are mixed with a stir bar, controlled by a magnet located below the cell. The contents of the cell are projected onto a video monitor using a camera (Lenox Instrument Company, Model STC-N63CJ) coupled to a borescope (Gradient Lens Corporation, Model HAWKEYE Pro) placed against the sapphire window. For both VLE and density measurements the sealed, but empty, view cell is flushed with propylene, at least three times, prior to loading the mixture components to remove entrapped air. At this point the residual air in the cell, which is at 0.1 MPa, is reduced to less than 10 ppm and the residual propylene is 0.032 g. First, 1.6 to 9.5 g of liquid hexane are loaded with a standard uncertainty of 0.001 g using a syringe and then 0.8 to 7.3 g of propylene are loaded with a standard uncertainty of 0.001 g using a high-pressure, transfer cylinder. After loading propylene, the valve on the view cell is closed and the transfer cylinder is chilled. This method reduces the vapor pressure of propylene in the transfer line and cylinder that minimizes the loss of propylene

are u(T )/K = 0.22 + 10−6(75 + (T /K − 273.15))2 and u(p) = 0.07 MPa, respectively. The expanded accumulated uncertainty in the reported mixture densities is Uc(ρ) = 0.8% with a coverage factor, k = 2, which corresponds to a confidence interval of approximately 95%. B



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3. RESULTS AND DISCUSSION 3.1. Phase Equilibrium. Table 2 lists the experimental BP, DP, and CP data determined in this study for six mixtures with propylene mole fractions of 0.1414, 0.4466, 0.6060, 0.6127, Table 2. Experimental Bubble, Dew, and Critical Point Data for the Propylene (1) + Hexane (2) System Determined in This Studya T/K 295.6 324.6 350.8 378.4 425.0 453.5 464.0 295.3 309.7 324.5 324.5 324.5 349.2 375.6 425.1 448.1 466.9 468.4 295.6 324.7 372.2 374.2 295.3 323.1 326.3 355.2 394.3 423.3 435.1 295.9 325.5 314.9 305.3 294.8 295.1 309.8 325.1 350.5 370.9 371.1 376.9 382.1 387.1

p/MPa x1 = 0.1414 0.22 0.39 0.63 0.91 1.79 2.37 2.66 x1 = 0.4466 0.46 0.66 0.92 0.92 0.92 1.37 2.04 3.57 4.50 5.18 4.97 x1 = 0.6060 0.69 1.25 2.71 2.78 x1 = 0.6127 0.75 1.22 1.29 2.09 3.61 4.96 5.32 x1 = 0.8985 1.04 1.94 1.63 1.26 x1 = 0.9090 1.03 1.01 1.39 1.96 3.07 4.31 4.29 4.61 4.98 5.20

transition BP BP BP BP BP BP BP

Figure 2. Isopleths for the propylene + hexane system at propylene mole fractions of 0.1414 (▽), 0.4466 (□), 0.6127 (△), 0.6060 (◇), 0.8985 (▷), 0.9003 (○). The filled symbols are experimental mixture critical points determined in this study. Lines are drawn to guide the eye.

0.8985, 0.9090 from temperatures of 290 to 470 K. Figure 2 shows the p−T isopleths for the propylene + hexane system obtained in this study. Note that only four p−T isopleths are found in Figure 2 because the results from independent experiments at two propylene mole fractions of ∼0.6 and ∼0.9 superimpose, confirming the reproducibility of the techniques used in this study. 3.2. Propylene + Hexane Mixture Densities. Tables 3 to 7 list experimental propylene + hexane mixture density data for five mixture compositions. The densities are determined from 295 to 584 K and pressures to 68 MPa. Figure 3 shows propylene + hexane mixture densities as a function of pressure. As seen in Figure 3, mixture densities increase with an increase in pressure and decrease with an increase in temperature. Figure 3d shows the high degree of reproducibility of independent density measurements from data for mixtures with propylene mole fraction of 0.9003 and 0.8985 at 295 and 324 K. These two molar compositions only differ by 0.1%. To the best of our knowledge, no literature density data are available for the propylene + hexane system to compare to the mixture densities determined in the present study. 3.3. Modified Tait Equation. The experimental propylene + hexane mixture density data are correlated with the modified Tait eq 1, which allows for the interpolation of mixture densities at any given T and p ρ − ρ0 p+B = C log10 ρ p0 + B (1)

BP BP BP BP BP BP BP BP BP CP DP BP BP BP BP BP BP BP BP BP BP BP BP BP BP BP

where p0 = 0.1 MPa, ρ0 is the temperature-dependent density at p0, B is a temperature-dependent parameter, and C is a temperature-independent constant approximately equal to 0.200 for hydrocarbons and simple hydrocarbon mixtures.16,17 Both ρ0 and B are fitted to individual isotherm-isopleth density data by minimizing the average absolute deviation (AAD) between experimental, Xi,exp, and calculated, Xi,cal, densities

BP BP BP BP BP BP BP BP BP CP

AAD/% = 100 ×

1 N

N

∑ i=1

Xi ,exp − Xi ,cal Xi ,exp

(2)

where N is the total number of data points. The parameters ρ0 and B are then fit to a quadratic function of temperature as given in eqs 3 and 4, respectively

a

Bubble point, BP; dew point, DP; and critical point, CP. Standard uncertainties, u, are u(x 1 ) = 0.0007, u(T)/K =

2

ρ0 =

0.22 + 10−6(75 + (T /K − 273.15))2 , u(p) = 0.07 MPa

∑ aiT i i=0

C

(3) DOI: 10.1021/acs.jced.6b00181 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

Table 3. Experimental Density, ρ, for Propylene (1) + Hexane (2) Mixtures at x1 = 0.1414a T/K 295.6

324.6 ρ/kg·m

p/MPa 0.7 1.3 3.5 8.2 10.7 14.9 20.7 24.0 27.2 30.8 37.1 42.7 47.8 48.0 48.2 48.3 52.0 55.3 55.6 58.0 61.9 66.1 66.2

−3

ρ/kg·m

p/MPa

638.3 639.2 642.0 646.8 649.7 653.6 659.5 662.5 665.6 668.6 673.8 678.5 682.2 682.8 682.8 682.8 685.5 688.2 688.2 689.8 692.5 695.3 695.3

425.0

0.7 1.2 1.9 3.6 5.0 7.1 8.9 9.6 10.3 12.3 13.6 17.1 19.7 24.1 27.4 30.7 34.3 37.5 44.3 47.8 51.3 55.2 58.4 62.7 66.5

−3

ρ/kg·m

p/MPa

611.8 612.7 614.0 616.6 618.8 621.9 624.1 625.5 625.9 628.6 630.0 634.1 637.3 642.0 644.9 648.2 651.6 654.6 660.5 663.5 666.6 669.7 672.2 675.4 678.5

500.9

1.8 3.7 5.1 7.0 10.4 17.3 17.5 20.6 24.1 27.6 30.9 34.3 37.8 41.1 44.6 51.9 54.4 58.6 61.8 63.6 66.7

−3

ρ/kg·m

p/MPa

494.2 504.2 510.4 517.7 528.7 546.6 547.3 553.9 561.4 567.6 573.2 578.9 584.7 589.1 594.3 603.8 607.1 612.3 615.7 617.9 621.0

584.6

12.4 13.9 20.8 24.6 27.5 31.0 37.3 37.9 41.4 44.7 44.8 47.1 48.1 51.5 54.9 58.4 61.8 64.6 67.4

−3

449.6 457.6 486.8 500.5 508.9 517.7 531.3 531.3 537.5 543.2 543.5 549.0 549.0 554.6 559.9 565.4 569.8 574.0 577.4

p/MPa

ρ/kg·m−3

29.0 31.1 34.7 37.9 41.4 44.8 45.1 48.1 51.7 55.1 58.4 61.8 64.2 67.4

449.8 456.9 468.2 478.0 487.3 495.6 496.5 503.4 510.7 517.4 523.9 529.0 532.9 537.5

Standard uncertainties, u, are u(x1) = 0.0007, u(T)/K = 0.22 + 10−6(75 + (T /K − 273.15))2 , u(p) = 0.07 MPa and the combined experimental density uncertainty, Uc, is Uc(ρ) = 0.8% (k = 2, with 95% confidence level). a

Table 4. Experimental Density, ρ, for Propylene (1) + Hexane (2) Mixtures at x1 = 0.4466a

2

B=

324.4 −3

p/MPa ρ/kg·m

p/MPa ρ/kg·m

424.7 −3

(4)

i=0

T/K 295.3

∑ biT i

Reported here are the maximum deviation (Dmax) and bias for the fit of the density data as defined by eqs 5 and 6, respectively

501.0 −3

p/MPa ρ/kg·m

−3

p/MPa ρ/kg·m

0.5

592.6

0.9

560.3

4.8

433.6

7.9

293.3

0.7

594.3

1.4

561.7

7.2

448.4

9.7

331.0

1.4

595.1

1.9

562.5

10.7

467.6

10.1

330.6

2.1

596.8

2.1

563.2

17.4

492.5

12.6

363.8

2.8

597.6

3.4

565.5

21.0

502.4

17.2

401.7

3.7

598.9

4.9

568.9

24.3

511.5

21.9

428.7

4.8

600.6

6.8

571.9

31.0

525.0

22.0

433.3

6.8

603.6

10.2

577.8

37.8

537.6

24.0

436.5

10.2

607.9

17.2

588.5

44.7

548.4

24.4

440.8

16.9

615.3

23.8

597.6

51.5

558.4

31.2

463.6

24.1

622.5

30.7

606.1

58.3

567.4

34.6

473.3

30.9

629.0

37.8

614.0

63.9

573.9

37.9

482.2

38.1

634.6

44.7

620.3

67.5

578.1

45.0

497.7

44.5

640.3

51.4

626.2

51.6

509.9

51.6

646.2

58.3

631.8

57.9

519.9

58.3

651.1

63.7

636.0

63.2

527.6

63.0

654.6

67.5

638.9

67.2

533.2

66.5

657.1

⎛ X i ,exp − Xi ,cal Dmax /% = max⎜⎜ Xi ,exp ⎝ Bias/% = 100 ×

1 N

N

⎛ Xi ,exp − Xi ,cal ⎞ ⎟⎟ Xi ,exp ⎝ ⎠

∑ ⎜⎜ i=1

⎞ ⎟ ⎟ ⎠

(5)

(6)

Table 8 lists the temperature range, pressure range, and mixture compositions for the six parameters obtained from the best fit of eqs 1, 3, and 4 with a constant value for C = 0.202. The AADs for every mixture composition are within the accumulated experimental uncertainty of the density data. Mixture densities below 0.450 g/cm3 are not included in the fit of the Tait equation because these data exhibit a greater than 1.0% deviation from the fit. Hence, the Tait parameters for reported isopleths are applicable to the restricted temperature ranges given in Table 8 to avoid fitting densities outside the applicable range for this equation. Figure 4 shows the deviation of experimental propylene + hexane mixture densities from the Tait equation calculated densities for four compositions. The deviations are generally within 0.5%, which is within the expanded accumulated experimental uncertainty of the data.

a

Standard uncertainties, u, are u(x 1 ) = 0.0007, u(T)/K = 0.22 + 10−6(75 + (T /K − 273.15))2 , u(p) = 0.07 MPa and the combined experimental density uncertainty, Uc, is Uc(ρ) = 0.8% (k = 2, with 95% confidence level). D



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Table 5. Experimental Density, ρ, for Propylene (1) + Hexane (2) Mixtures at x1 = 0.6060a T/K 295.0 p/MPa 1.4 5.2 6.8 10.3 17.2 23.9 30.6 37.5 44.5 51.1 57.9 63.1 66.8

324.6 ρ/kg·m

−3

568.0 574.6 577.0 582.2 592.0 600.0 606.9 613.5 619.4 624.9 630.0 634.3 636.7

374.1 ρ/kg·m

p/MPa 1.3 1.8 2.5 3.4 5.3 6.8 10.3 17.4 23.9 30.7 37.9 44.4 51.4 58.1 63.0 67.2

−3

533.2 534.5 536.2 538.3 542.5 545.6 552.4 564.6 573.8 583.4 592.4 599.1 606.4 612.2 616.2 619.4

424.0 ρ/kg·m

p/MPa 2.8 2.9 3.8 5.5 7.3 10.6 17.4 24.0 31.0 37.9 38.0 44.7 51.3 58.1 63.3 66.9

−3

ρ/kg·m

p/MPa

465.7 467.0 471.2 479.0 486.2 497.9 515.3 528.5 540.7 551.3 551.7 560.9 569.2 577.4 583.4 587.4

501.0

10.1 17.5 24.3 30.8 37.9 44.7 51.4 58.3 63.0 66.9

−3

p/MPa

ρ/kg·m−3

9.2 10.7 12.4 17.3 20.7 25.2 29.0 31.0 37.9 44.8 51.4 58.4 63.0 67.0

230.6 272.7 306.1 362.0 386.4 411.3 429.0 435.7 455.7 473.6 485.9 500.0 506.6 512.8

427.5 462.7 484.2 500.6 514.3 523.9 534.9 546.0 551.0 556.0

Standard uncertainties, u, are u(x1) = 0.0007, u(T)/K = 0.22 + 10−6(75 + (T /K − 273.15))2 , u(p) = 0.07 MPa and the combined experimental density uncertainty, Uc, is Uc(ρ) = 0.8% (k = 2, with 95% confidence level). a



T/K

T/K

295.6

a

294.9

325.5

p/MPa

ρ/kg·m−3

p/MPa

ρ/kg·m−3

1.7 1.9 2.6 3.6 4.9 7.0 8.5 10.1 10.5 13.8 16.7 17.2 23.2 29.1 34.4 38.1 41.0 44.8 48.2 51.6 54.9 58.4 61.8 67.3

516.2 518.5 518.5 522.1 524.5 528.5 532.3 535.7 535.7 541.2 546.4 546.4 555.2 562.8 568.4 573.9 576.4 581.2 584.2 588.0 590.4 594.2 596.6 601.4

2.4 2.8 3.5 4.7 6.8 10.7 13.8 16.1 20.9 27.6 34.3 41.3 48.2 51.6 55.0 58.4 61.8 64.2 67.1

471.9 474.2 476.7 481.2 488.5 500.7 508.3 513.5 523.6 535.1 544.8 554.1 561.7 566.7 569.5 574.2 576.8 580.1 582.4

325.1

370.2

423.6

p/MPa ρ/kg·m−3 p/MPa ρ/kg·m−3 p/MPa ρ/kg·m−3 p/MPa ρ/kg·m−3 1.2 1.8 2.8 3.3 4.4 10.3 17.1 24.1 30.8 37.8 44.2 51.2 55.0 58.3 67.3

a

515.1 516.8 519.0 520.8 523.3 535.5 547.5 557.1 565.9 573.2 580.2 587.4 591.5 593.8 601.9

2.1 2.8 3.7 7.0 10.4 16.5 24.0 30.9 37.5 44.5 51.5 58.3 63.7 67.2

470.3 473.5 476.8 489.5 499.8 515.1 530.6 542.4 551.5 559.6 567.6 574.5 580.2 583.3

5.2 5.9 7.3 10.7 14.2 17.3 20.8 24.1 31.3 38.0 44.6 46.9 48.0 50.6 51.4 57.8 61.8 67.1

385.2 392.1 406.0 429.7 447.3 459.2 469.1 478.3 495.2 508.5 519.7 523.6 523.6 529.1 529.1 538.5 544.3 550.3

16.6 17.4 20.8 24.3 31.2 37.9 44.8 51.6 58.3 59.4 63.7 67.1

385.2 389.1 408.2 424.0 449.1 462.3 479.8 493.0 504.1 504.8 512.3 516.8

Standard uncertainties, u, are u(x 1 ) = 0.0007, u(T)/K =

0.22 + 10−6(75 + (T /K − 273.15))2 , u(p) = 0.07 MPa and the combined experimental density uncertainty, Uc, is Uc(ρ) = 0.8% (k = 2, with 95% confidence level).

properties of mixtures over wide ranges of temperature, pressure, and composition. Therefore, in this study, four different EoS modelsincluding the SRK,5 the PR,6 the MSL,7,8 and the PC-SAFT9 equationare used to predict VLE and mixture densities for the propylene + hexane system. Only a brief description of the MSL and PC-SAFT equations are given here because details for the SRK and PR equations are found elsewhere.5,6 Standard van der Waals one-fluid mixing rules are used for the energetic parameter a in the MSL, SRK, and PR equations.

Standard uncertainties, u, are u(x 1 ) = 0.0007, u(T)/K =

0.22 + 10−6(75 + (T /K − 273.15))2 , u(p) = 0.07 MPa and the combined experimental density uncertainty, Uc, is Uc(ρ) = 0.8% (k = 2, with 95% confidence level).

4. MODELING The Tait equation is useful for interpolating mixture density data; however, it is not a robust tool for predicting the thermodynamic E



Article

Figure 3. Density isotherms for propylene (1) + hexane (2) mixtures obtained in this study. In all four panels, data are presented at ∼295 K (○), ∼325 K (□), ∼425 K (◇). Panels a, b, and c show data at ∼501 K (△). Panel a also shows data at ∼585 K ( × ) and panels c and d show data at ∼375 K (▽). Panel d shows data at ∼295 K (●) and ∼325 K (■) for x1 = 0.8985. Lines are drawn to guide the eye.

complex phase behavior of polydisperse polymer systems, where cubic EoSs perform poorly. The residual Helmholtz free energy for the MSL EoS is

Table 8. Best Fit Parameters of the Modified Tait Equation along with AAD, SD, and Dmax for Propylene (1) + Hexane (2) Mixture Densities at Four Compositions and with C = 0.202 x1 = Ta/K p/MPa N, data points fit a0/g·cm−3 a1/g·cm−3·K−1 103 × a2/g·cm−3·K−2 b0/MPa b1/MPa·K−1 104 × b2/MPa·K−2 AAD/% SD Dmax

0.1414 295 to 425 0.7 to 66.7 70 615.800 0.9448 −2.9560 170.200 −0.5472 3.782 0.10 0.11 0.45

0.4466 295 to 425 0.5 to 67.5 48 489.545 1.7695 −4.8000 334.251 −1.54875 18.000 0.09 0.09 0.35

0.6060 295 to 375 1.3 to 67.5 45 353.855 2.4278 −5.8000 135.030 −0.4533 2.825 0.23 0.22 0.67

⎛v − b⎞ ar a − aig d a ⎟ − = = (v − b)ln⎜ +d ⎝ v ⎠ vRT RT RT b

0.9003 295 to 370 1.2 to 67.3b 44 −973.400 11.2300 −2.0990 321.978 −1.1070 22.640 0.20 0.10 0.60

(7)

The expression for pressure can be obtained by differentiating the Helmholtz energy with respect to volume p 1−d d ⎛v − b⎞ a ⎟ − = − ln⎜ 2 ⎝ ⎠ RT v b v v RT

(8)

The molar volume is calculated using the Péneloux volume translation vt = v + c (9) The MSL EoS has four mixture parameters, a, b, c, and d, that are calculated using linear and quadratic mixing rules as given by Krenz et al.7 Table 9 lists pure component properties used with the SRK, PR, and MSL EoSs. 4.2. PC-SAFT EoS. The residual Helmholtz free energy, ar, obtained with the PC-SAFT EoS is given by9

a

Experimental mixture density data are reported beyond the highest temperature listed in this table; however, those high temperature data are not used for the fit. bThree data points below 10 MPa at T = 370 K are excluded from the fit.

4.1. MSL EoS. The MSL equation is Neau’s version of the Sanchez−Lacombe equation modified to include a Pénelouxtype volume translation.7 Neau obtained the residual Helmholtz free energy by integrating the pressure expression from the original Sanchez−Lacombe equation. Although the simplicity, speed of calculations, and robustness of the MSL equation is similar to that exhibited by standard cubic equations of state, the MSL equation has been shown to accurately predict the

a r = a hs + achain + adisp hs

(10)

chain

where a and a are the contributions from hard-sphere, segment−segment interactions and from chain formation, respectively. Here, we ignore any polar contributions from propylene to the residual Helmholtz free energy. Table 10 lists the pure component number of segments, m, temperatureindependent segment diameter, σ, and the interaction energy, F



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Figure 4. Deviation plots for propylene (1) + hexane (2) mixture density data obtained in this study from the fit of the modified Tait equation at ∼295 K (○), ∼ 325 K (□), ∼ 425 K (△), and ∼375 K (▽).

Table 9. Pure Component Critical Temperature, Tc, Critical Pressure, pc, and Acentric Factor, ω, Used with the SRK, PR, and MSL Equations18 component

ω

Tc/K

pc/MPa

propylene hexane

0.144 0.300

364.9 507.6

4.6 3.0

experimental VLE data determined in this study. Note, however, the PC-SAFT EoS provides a better estimate of the mixture critical point compared to predictions with the MSL EoS. Nevertheless, both equations predict type I phase behavior with a continuous critical mixture curve.19,20 Although not shown here, the phase boundary curves calculated with the SRK and the PR EoS are essentially indistinguishable from the phase boundary curves calculated with the PC-SAFT and MSL EoS. These results demonstrate that the VLE of the propylene + hexane system can accurately be predicted using simple cubic equations of state. Figure 6 shows the percent deviation for experimental to predicted propylene + hexane BP pressures using two different values for kij. Figure 6a shows the PC-SAFT results and Figure 6b shows the MSL results. Deviations for the BP pressure primarily fall to one side of zero when using kij = 0.000, with a maximum deviation of ∼36%. Note, however, the deviations are randomly spread around zero when using kij = 0.028, with a maximum deviation of ∼20%. Overall, with kij = 0.028, 90% of the data fall within ±10% of the PC-SAFT predicted pressures. With kij = 0.028, the BP pressure AAD is 4.7 and 4.4 for PC-SAFT and MSL, respectively. The results suggest the importance of the binary interaction parameter, kij, to accurately predict phase boundary curves. Although the results are not shown here, the BP pressure deviations calculated with the SRK and the PR EoS are similar to those calculated with the PC-SAFT and MSL EoS. Figure 7 shows the percent deviation for experimental to predicted propylene + hexane BP temperatures using two different kij values. Figure 7a shows PC-SAFT results and Figure 7b shows MSL results. With both models the deviations mostly fall on one side of zero when using kij = 0.000, with a maximum deviation of ∼8%. The deviations are randomly spread around

Table 10. PC-SAFT Pure Component Properties Obtained from Gross and Sadowski.9 component

M/g·mol−1

m

σ/Å

ε·kB−1/K

propylene hexane

42.081 86.177

1.9597 3.0576

3.5356 3.7983

207.19 236.77

ε/kB, required for PC-SAFT calculations for nonassociating molecules. The mixing rules for the three pure component parameters are given by eqs 11 to 13 N

m=

σij =

∑ ximi i=1

(11)

1 (σi + σj) 2

(12)

εij = (1 − kij) εiεj

(13)

4.3. Modeling Results. 4.3.1. Phase Behavior Calculations. Figure 5 compares experimental phase boundary curves for the propylene + hexane system with PC-SAFT and MSL EoS calculated phase envelopes. When the binary interaction parameter, kij, is set to zero both models under predict bubble point data. However, when a small positive value for kij of 0.028 is used, both models provide a good representation of the G



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Figure 5. Phase boundary curves for the propylene (1) + hexane (2) system with propylene mole fractions of 0.1414 (▽), 0.4466 (□), 0.6127 (△), 0.6060 (◇), 0.8985 (▷), and 0.9003 (○). The filled symbols are experimental mixture critical points determined in this study. Solid lines are model calculations with kij = 0.028 and dashed lines are model calculations with kij = 0.000; (a) PC-SAFT EoS and (b) MSL EoS.

Figure 6. Deviation of experimental bubble point pressures at temperatures from 295 to 470 K from predicted bubble point pressures using two different kij values for all four isopleths. (a) PC-SAFT equation and (b) MSL equation, both with kij = 0.000 (○) and kij = 0.028 (●).

Figure 7. Deviation of experimental bubble point temperatures from ∼0.5 to ∼5.5 MPa from predicted bubble point temperatures using two different kij values for all four isopleths. (a) PC-SAFT equation and (b) MSL equation, both with kij = 0.000 (○) and kij = 0.028 (●).

zero when using kij = 0.028, with a maximum deviation of ∼5%. Overall, with kij = 0.028, 99% of the data fall within ±3% of the predicted temperatures. With kij = 0.028, the AAD of the BP temperature is 0.9 and 0.7% for calculations with the PC-SAFT and MSL EoS, respectively, with the largest errors occurring at pressures less than 1.0 MPa. Although the results are not shown here, the BP temperature deviations calculated with the SRK and the PR EoS are similar to those calculated with PC-SAFT and MSL EoS.

4.3.2. Mixture Density Calculations. Propylene + hexane mixture densities are also calculated with the SRK, PR, MSL, and PC-SAFT equations over the entire experimental T, p, and x range. Table 11 lists the AAD, SD, Dmax, and bias for the density predictions with all four models when kij is set to zero. The AADs range from 3% to 9% and maximum deviations are up to 20%. These results show that PC-SAFT provides better density predictions compared to the SRK, PR, and MSL equations. In fact, the performance of the MSL EoS is poor compared to the H



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Table 11. Absolute Average Deviation, Standard Deviation, Maximum Deviation, and Bias for the Predictions of Propylene + Hexane Densities over Entire Experimental T, p, and x Range Using Different Equations of State and with kij = 0.000a EoS

AAD/%

SD/%

Dmax/%

bias/%

PC-SAFT SRK PR MSL

3.40 4.33 6.86 9.40

1.66 2.86 3.06 5.19

8.52 13.02 11.67 20.37

−3.18 4.77 −6.42 −7.38

Table 13. Absolute Average Deviation, Standard Deviation, Maximum Deviation, and Bias for the Predictions of Propylene + Hexane Mixture Density Predictions Using PC-SAFT with kij = 0.028a x1

T/K

no. of data points

AAD/%

SD/%

Dmax/%

bias/%

0.000b

295.0 400.0 500.0 295.6 324.7 425.0 501.0 584.6 295.4 324.5 424.7 501.1 295.0 324.7 374.2 424.1 501.1 295.7 325.6 295.0 370.2 325.1 423.6 295.0 400.0 500.0

14 14 15 23 26 21 18 14 18 17 13 14 13 16 16 10 14 24 19 15 18 14 12 14 15 15

0.34 0.70 1.32 0.92 0.69 1.10 1.21 0.46 2.96 3.27 3.18 2.54 4.05 3.93 3.99 3.64 2.19 4.28 3.90 4.32 3.82 3.78 2.32 1.05 1.41 0.96

0.20 0.42 1.13 0.26 0.34 0.34 0.57 0.32 0.44 0.22 0.79 0.99 0.21 0.13 0.55 1.05 1.16 0.21 0.26 0.13 0.78 0.14 1.33 0.27 0.86 0.46

0.66 1.30 4.92 1.21 1.19 1.41 1.80 1.03 3.89 3.71 3.89 3.88 4.36 4.17 4.49 4.46 3.68 4.63 4.28 4.53 4.60 3.99 3.58 1.43 3.70 1.59

−0.03 −0.69 −0.99 −0.92 −0.69 −1.10 −1.11 −0.14 −2.96 −3.27 −3.18 −2.54 −4.05 −3.93 −3.99 −3.64 −2.19 −4.28 −3.90 −4.32 −3.82 −3.78 −2.20 1.05 1.41 0.96

0.1414

a

Absolute average deviation, AAD; standard deviation, SD; maximum deviation, Dmax. The total number of fitted data points is 335. 0.4466

Table 12. Absolute Average Deviation, Standard Deviation, Maximum Deviation, and Bias for the Predictions of Propylene + Hexane Densities over Entire Experimental T, p, and x Range Using Different Equations of State and with kij = 0.028a EoS

AAD/%

SD/%

Dmax/%

bias/%

PC-SAFT SRK PR MSL

2.73 4.62 6.53 9.32

1.46 2.91 3.07 5.14

4.63 13.38 11.42 20.28

−2.38 5.12 −5.98 −7.17

0.6060

0.8985 0.9003

a

Absolute average deviation, AAD; standard deviation, SD; maximum deviation, Dmax. The total number of fitted data points is 335.

1.000b

performance of the SRK and PR equations. Table 12 lists the AAD, SD, Dmax, and bias for the density predictions with all four models when kij is set to 0.028. In this instance, the AADs and maximum deviations are similar to those observed when kij equals zero. These results suggest that density calculations are relatively insensitive to small changes in the value of kij regardless of which of the four equations are used. The results for the PC-SAFT density calculations are scrutinized to ascertain any trends in the model calculations with respect to mixture composition and temperature. Table 13 lists the AAD, SD, Dmax, and bias for PC-SAFT calculations of propylene + hexane mixture densities at specific compositions and temperatures and with kij = 0.028. Table 13 also lists the AAD, SD, Dmax, and bias for PC-SAFT calculations of pure propylene and pure hexane densities for pressures 5 to 70 MPa are obtained from the NIST WebBook.14 The AAD for pure propylene densities range from ∼1.0 to 1.5% and the AAD for pure hexane densities range from 0.3 to 1.3%. Note that the AAD for mixtures with a propylene mole fraction of 0.1414 is ∼1.0% for all the investigated temperatures whereas the AAD increases to ∼3.0 to 4.5% for the other three mixtures with higher propylene mole fractions of 0.4466, 06060, and 0.9003. This increase in AAD for propylene-rich mixtures is a bit surprising given that calculations of the densities of each pure component exhibit AAD values lower than 1.5%. It is also evident, based on the negative bias values, that the PC-SAFT consistently overpredicts mixture densities regardless of the amount of propylene in the mixture with the exception of the densities for pure propylene.

a

Absolute average deviation, AAD; standard deviation, SD; maximum deviation, Dmax. The total number of fitted data points is 335. b Propylene and hexane density data for pressures 5 to 70 MPa are retrieved from the NIST WebBook.14

representation of the reported high-temperature, high-pressure density data although equivalent predictions of the reported VLE data are obtained with the SRK, PR, and MSL equations. The VLE calculations performed with each EoS are sensitive to small changes to the binary interaction parameter, kij, whereas the density calculations are relatively insensitive to the same changes in the value of kij.

■

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b00181. GC−MS analysis of the hexane used in this study. (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1 (804) 827 7031. Fax: +1 (804) 828 3846. Notes

The authors declare no competing financial interest.

■

5. CONCLUSIONS VLE data for propylene + hexane mixtures are reported at high temperatures and modest pressures of 5.5 MPa together with mixture density data also at high temperatures but at pressures to 70 MPa. These data complement previously reported data for ethylene + hexane mixtures. The PC-SAFT EoS provides the best

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Phase Behavior and Densities of Propylene+ Hexane Binary Mixtures

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